FISHERY BULLETIN: VOL. 87. NO. 2, 1989 



where M is the instantaneous natural mortality rate. 

 (The dot subscript indicates a summation over all 

 j, i.e., 



NV, = 1 NV,, 



and similarly 



NV , = 1 NVt , 



A 



E 



(=1 



where X is the maximum age attained.) The mean 

 length-at-age L, can be described by the von Berta- 

 lannfy growth curve: 



L, = L„{1 - exp[-K{t - t,)]]. 



(2) 



It is assumed that lengths-at-age are normally dis- 

 tributed around their mean, and that standard devia- 

 tions vary with age in the manner proposed by 

 Fournier and Breen (1983): 



SD, = a + 



b\ft 



(3) 



where a and 6 are constants. If lengths-at-age are 

 normally distributed around their mean in the un- 

 fished population, the probability that an individual 

 in cohort t will be found in length interval j is 



"y+i 



Qt, = 



SDt\/2~fi 



/ exp[-(fe - L,fl2 SDf] dh. 



(4) 



This was evaluated numerically. 



Equations (2) to (4) could be used to describe a con- 

 tinuous growth process. For computational tract- 

 ability, this model evaluates these relations only at 

 integer values of t. This is equivalent to assuming 

 that growth occurs yearly in one instantaneous in- 

 crement; or alternatively that all fishing mortality 

 occurs instantaneously when t assumes an integer 

 value. 



The number of individuals in all cohorts of the un- 

 fished population in length interval j is 



NV,,j= lQ,,,JVy,,.. 



(5) 



Length-specific fecundity can be described by 



fj = c hj (6) 



where c and d are regression constants and fj is the 



number of eggs produced by a female of length hj. 

 If spawning is assumed to occur once annually when 

 t assumes an integer value, total egg production by 

 an equilibrium population of females not subjected 

 to fishing is described by 



E = X NV f 



-'-'max ^ '■jJj 



(7) 



where h„ is the length at first maturity. E^^ is 

 calculated with appropriate parameter values and 

 with a fixed arbitrary value for TVq. 



The model is now extended to include fishing mor- 

 tality acting on all individuals whose length is equal 

 to or greater than a minimum legal size hj^. The 

 number of individuals at a particular size and age 

 in the fished population will be denoted by N, j. It 

 is assumed that no individuals are recruited to the 

 fishery before age 1, so that 



A/'i.. = iV(, exp(-M). 



(8) 



In this and all subsequent cohorts, the number of 

 individuals less than legal size will be the same as 

 in the virgin population and can be determined as 

 follows. The proportion of prerecruits in the virgin 

 population is given by 



QPR, 



1 



/ exp -[{h - L,fl2 son dh. 



SDt \J2 n 



(9) 



The number of prerecruits in each cohort is thus 

 NPR, . = QPR, NVt^ . (10) 



and the number of individuals exposed to the fishery 



is 



NR, 



N, . - NPR, 



(11) 



The overall survival rate of this cohort over one year 

 will be determined by natural mortality acting on 

 the prerecruits, and by both natural and fishing mor- 

 tality on the recruits: 



S, = [{NPR,JN,,)exp[-M]] 



+ [(NR,^JN,,)exp[-(F + M)]] (12) 



Beginning with cohort 1, the abundance of suc- 

 cessive cohorts can be determined: 



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